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A271527 a(n) = 1000^n + 1. 1

%I

%S 2,1001,1000001,1000000001,1000000000001,1000000000000001,

%T 1000000000000000001,1000000000000000000001,1000000000000000000000001,

%U 1000000000000000000000000001,1000000000000000000000000000001,1000000000000000000000000000000001

%N a(n) = 1000^n + 1.

%C All terms in this sequence are palindromes (A002113).

%C Also, A062395 written in base 2 (see example).

%C a(n) minus one gives the number of nodes at n-th level of a 1000-ary tree.

%C More generally, the ordinary generating function for sequences of the form k^n + m, is (1 + m - (1 + k*m)*x)/((1 - x)*(1 - k*x)), and the exponential generating function is exp(k*x) + m*exp(x).

%H Ilya Gutkovskiy, <a href="/A271527/a271527.pdf">Examples of the ordinary generating function for the sequences of the form k^n + m</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1001,-1000)

%F G.f.: (2 - 1001*x)/((1 - x)*(1 - 1000*x)).

%F E.g.f.: exp(1000*x) + exp(x).

%F a(n) = 1001*a(n-1) - 1000*a(n-2).

%F a(n) = A060365(n) + 1.

%F a(n) = A000533(3n), n>0.

%F a(n) = A007088(A062395(n)).

%F A007953(a(n)) = A007395(n).

%F A000035(a(n)) = A057427(n).

%F Sum_{n>=0} 1/a(n) = 0.501000001999002...

%F Lim_{n->infinity} a(n + 1)/a(n) = 1000.

%e a(n), n>0, is the binary representation of A062395(n)

%e n ------------------------------------------

%e 0 2........................................2

%e 1 1001.....................................9

%e 2 1000001.................................65

%e 3 1000000001.............................513

%e 4 1000000000001.........................4097

%e 5 1000000000000001.....................32769

%e 6 1000000000000000001.................262145

%e 7 1000000000000000000001.............2097153

%e 8 1000000000000000000000001.........16777217

%e 9 1000000000000000000000000001.....134217729

%t Table[1000^n + 1, {n, 0, 11}]

%t LinearRecurrence[{1001, -1000}, {2, 1001}, 12]

%o (PARI) x='x+O('x^99); Vec((2-1001*x)/((1-x)*(1-1000*x))) \\ _Altug Alkan_, Apr 09 2016

%o (Python)

%o for n in xrange(0,10**4):print(1000**n+1)

%o # _Soumil Mandal_, Apr 10 2016

%Y Cf. A000035, A000533, A007088, A007395, A007953, A057427, A060365, A062395, A152756.

%K nonn,base,easy

%O 0,1

%A _Ilya Gutkovskiy_, Apr 09 2016

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Last modified February 24 14:41 EST 2018. Contains 299623 sequences. (Running on oeis4.)